arXiv subjects

Sushant Kala

Publications and source records attributed to Sushant Kala.

7 recordsLinked to original sources

arXiv2026

An estimate for incomplete mixed character sums and applications

Let $q$ be a prime power and $m>1$ be any integer. Let $\mathbb F_{q^m}$ be the finite field of order $q^m$ and $\theta\in\mathbb F_{q^m}$ be such that $\mathbb F_{q^m} = \mathbb F(\theta)$. We obtain a nontrivial bound for the mixed character sum $\sum_{x \in\mathbb F}\chi(\theta+x)\psi(x)$, where $\chi$ and $\psi$ are multiplicative and additive characters of $\mathbb F_{q^m}$ and $\mathbb F$, respectively, using function field methods. As an application of our main result, we prove that for fixed $m$ and sufficiently large prime powers $q$, that satisfy certain conditions, $\mathbb F_{q^m}/\mathbb F$ possesses the weak line property for primitive normal elements. In particular, our result is a strengthening of existing results.

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arXiv2025

Lower bound for the canonical height on abelian varieties over totally p-adic extensions

Let $A/\mathbb{Q}$ be an abelian variety and let $\hat{h}$ be the canonical height on $A(\overline{\Q})$ associated to a symmetric ample line bundle $\mathcal{L}$ on $A$. We prove that $\hat h$ is bounded away from zero on non-torsion points of $A$ defined over the maximal totally $p$-adic extension of $\mathbb{Q}$, for all but finitely many primes $p$. More generally, for abelian varieties over a number field $K$, we obtain a similar height gap over certain infinite extensions of $K$, including Galois extensions with finite local degree at non-archimedean places.

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arXiv2025

A p-adic criterion for Lehmer's conjecture

For a non-zero algebraic number $\alpha$ of degree $d$, let $h(\alpha)$ denote its logarithmic Weil height. It is known that when $h(\alpha)$ is small, and $d$ is large, the conjugates of $\alpha$ are clustered near the unit circle and have angular equidistribution in the complex plane about the origin. In this paper, we establish a $p$-adic analogue of this result by obtaining lower bounds for $h(\alpha)$ in terms of the number of its conjugates that lie in a finite extension of $\mathbb{Q}_p$, for some prime $p$. As a consequence, we prove Lehmer's conjecture for all $\alpha$ such that $\gg \sqrt{d\log d}$ many of its conjugates lie in a finite extension of $\mathbb{Q}_p$.

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arXiv2025

On the distribution of shapes of pure quartic number fields

The shape of a number field is a subtle arithmetic invariant arising from the geometry of numbers. It is defined as the equivalence class of the lattice of integers with respect to linear operations that are composites of rotations, reflections, and positive scalar dilations. For a number field of degree $n$, the shape is a point in the space of shapes $\mathscr{S}_{n-1}$, which is the double quotient $GL_{n-1}(\mathbb{Z}) \backslash GL_{n-1}(\mathbb{R}) / GO_{n-1}(\mathbb{R})$. In this paper, we investigate the distribution of shapes in the family of pure quartic fields $K_m = \mathbb{Q}(\sqrt[4]{m})$. We prove that the shape of $K_m$ lies on one of ten explicitly described torus orbits in $\mathscr{S}_3$, determined by the sign and residue class of $m \bmod 32$. It is shown that the shape on a given torus orbit is completely determined by two parameters, one of which varies continuously, while the other takes values in a discrete set. As a result, the distribution of shapes in this family is governed by a product of a continuous and a discrete measure. Our results shed new light on a question posed by Manjul Bhargava and Piper H concerning the distribution of shapes in families of non-generic number fields of fixed degree. Notably, the limiting distribution in our case does not arise as the restriction of the natural measure on $\mathscr{S}_3$ induced by Haar measure on $GL_3(\mathbb{R})$.

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arXiv2024

On points of small height in infinite extensions

In this paper, we introduce the notion of asymptotically positive infinite extensions of $\mathbb{Q}$, in the spirit of the Tsfasman-Vl\u{a}du\c{t} theory of asymptotically exact families of number fields. For asymptotically positive extensions, we obtain lower bounds on the logarithmic Weil height, establishing the Bogomolov property for a wide range of infinite non-Galois extensions. Our result encompasses the famous theorem of E. Bombieri and U. Zannier on Bogomolov property for totally $p$-adic extensions of type $(e,f)$. Additionally, our theorem can be interpreted as a $p$-adic equidistribution result on conjugates of $\alpha$, resonating with the archimedean equidistribution theorem \`{a} la F. Amoroso-M. Mignotte and Y. Bilu. In the parallel setting of elliptic curves, we derive lower bounds on the canonical height for points on an elliptic curve over asymptotically positive extensions, without any restriction on its reduction type. In particular, this extends a result of M. Baker in the context of totally $\nu$-adic extensions, where the elliptic curve is assumed to have semistable reduction at $\nu$.

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arXiv2024

On the lowest zero of Dedekind zeta function

Let $\zeta_K(s)$ denote the Dedekind zeta-function associated to a number field $K$. In this paper, we give an effective upper bound for the height of first non-trivial zero other than $1/2$ of $\zeta_K(s)$ under the generalized Riemann hypothesis. This is a refinement of the earlier bound obtained by Omar Sami.

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arXiv2023

Lower bound on height of algebraic numbers and low lying zeros of the Dedekind zeta-function

In this paper, we establish lower bounds on Weil height of algebraic integers in terms of the low lying zeros of the Dedekind zeta-function. As a result, we prove Lehmer's conjecture for certain infinite non-Galois extensions conditional on GRH. We also introduce and study a condition on prime ideals with small norms for arbitrary infinite extensions, in the spirit of a prime splitting condition for infinite Galois extensions introduced by E. Bombieri and U. Zannier.

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